The mechanism
Feedback compares observed state with a target and changes the next control command according to that error. A velocity-input model assumes a lower-level system can achieve the commanded velocity; it does not directly model forces, torques or actuator dynamics. Proportional control commands velocity in proportion to position error. The gain and sample interval jointly affect a discrete implementation. For an ideal sampled integrator with no delay or saturation, the error recurrence can be derived exactly from its update rule. Its convergence condition must not be generalized to a delayed or mechanically flexible robot. The example exposes that dependence with numbers: a gain that reduces error at one sampling rate can alternate and grow at another effective gain.
- A target and measured position define error at the start of each sample.
- The proportional controller maps that error into a commanded velocity with a gain measured per second.
- The ideal plant integrates that velocity for the sample duration before the next observation.
- The resulting recurrence is checked for convergence within the declared no-delay, unsaturated model.
Science and assumptions
Feedback closes the action loop · explicit relationship
In this ideal sampled velocity-input model, error is multiplied by one minus gain times sample duration; convergence requires that multiplier’s magnitude be below one.
Symbols and units
- eₙ
- Target minus current position at sample n
m
- K
- Proportional velocity gain
1/s
- Δt
- Positive fixed sample duration
s
Let the target be 1 m, initial position 0 m, sample duration 0.1 s and gain 2/s. The error multiplier is 1−2×0.1=0.8, so the first errors are 1, 0.8 and 0.64 m. With gain 24/s at the same interval, the multiplier becomes −1.4 and the errors are 1, −1.4 and 1.96 m: they alternate and grow. This ideal discrete example has no velocity cap, noise or delay. A real actuator may saturate before those commands are achieved, changing the dynamics and requiring a different analysis.
Assumptions Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.
Engineering example
Let the target be 1 m, initial position 0 m, sample duration 0.1 s and gain 2/s. The error multiplier is 1−2×0.1=0.8, so the first errors are 1, 0.8 and 0.64 m. With gain 24/s at the same interval, the multiplier becomes −1.4 and the errors are 1, −1.4 and 1.96 m: they alternate and grow. This ideal discrete example has no velocity cap, noise or delay. A real actuator may saturate before those commands are achieved, changing the dynamics and requiring a different analysis.
Distinctions and tradeoffs
A positive proportional gain alone does not establish stability for every sampled or delayed physical system.
Larger gain can reduce error faster within a valid model but can amplify noise, hit limits or destabilize sampled/delayed behavior.
Robotic action depends on control rate and physical feedback, beyond the semantic choice of a target.
Specifications and interfaces
- Mechanism boundary
- How does measured error change the next command?
Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.
- Evidence type
- Authored worked example
Worked explanation with a bounded numeric or state example. This lesson does not claim an executing implementation of an external model or platform.
Interfaces
- Declared input and state Input
- A target and measured position define error at the start of each sample.
- Checked outcome Output
- Derive and evaluate a sampled position-error recurrence under an explicit velocity-input model.
Check your understanding
For the ideal sampled model, what happens when KΔt = 2.4?
- Error shrinks monotonically
- Error alternates and grows because the multiplier is −1.4
- Error is always zero
Answer and reasoning
Error alternates and grows because the multiplier is −1.4
The multiplier has magnitude greater than one, so the ideal recurrence grows while changing sign.
Evidence and scope
This is a worked explanation, not an executing external platform, model or robot. The quantities and fixtures are authored. Source scope is recorded in the automation learning review manifest; no source benchmark or private system behavior is claimed. Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.
The worked example explains this mechanism. General numerical labs are a reference; they do not execute this lesson’s full mechanism.